Nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity

In this research, we constructed the exact travelling and solitary wave solutions of the Kudryashov–Sinelshchikov (KS) equation by implementing the modified mathematical method. The KS equation describe the phenomena of pressure waves in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity. Our new obtained solutions in the shape of hyperbolic, trigonometric, elliptic functions including dark, bright, singular, combined, kink wave solitons, travelling wave, solitary wave and periodic wave. We showed the physical interpretation of obtained solutions by three-dimensional graphically. These new constructed solutions play vital role in mathematical physics, optical fiber, plasma physics and other various branches of applied sciences.

This research work is arranged as follows, we explain the introduction in Section 1. We describe the proposed technique in Section 2. We implemented the described technique on KS equation and found the exact travelling and solitary solutions in Section 3. This work end at conclusion in Section 4.

Description of proposed method
Here we describe the main future of the modified mathematical technique for finding the solutions of nonlinear partial differential equations (PDEs). We consider the general form of nonlinear PDEs as Where Q denote to polynomial function of u(x, t) and their derivatives. We explain the features of modified mathematical technique as Step 1. We consider the transformations of travelling wave as In Equation (6), λ is the wave frequency. We obtain the ODE of Equation (5), as In Equation (7), R denote to polynomial function in U(ζ ) and their derivative.
Step 2. We consider the trial solution of Equation (7), as Here (a i , b i , c i , d i ) are constants which calculate later, the derivatives of (ζ ) satisfy the following auxiliary equation In Equation (9), β , i s are real constants which are found later.
Step 3. We balance the terms of nonlinear and derivative of higher order in Equation (7), determined N of Equation (8).
Step 4. Putting Equation (8) in Equation (9) and collecting every coefficients of j (ζ ) i (ζ )(i = 1, 2, 3, . . . N; j = 0, 1), then every coefficients make zero and get a system of equation, solve these system of equations using any computer software, the values of these parameters (a i , b i , c i , d i ), are found.
Step 5. Substituting parameters values which are obtained and (ζ ) in Equation (9), then we obtain the required solutions of Equation (5).

KS equation
Here we apply the described technique to construct the solitary wave solutions for the KS equation.
We apply transformation of the wave Substituting Equation (11) in Equation (10) and integrating once w.r to ζ with zero integration constant, we obtain the following: We balance the term of nonlinear and derivative of higher order in Equation (12), we get N = 2. Trial solution of Equation (12), take as Substituting Equation (13) in Equation (12) and collect every coefficients of j (ζ ) i (ζ )(i = 1, 2, 3, . . . N; j = 0, 1), compare every coefficients to zero. We obtain a system of equations. These system of equations solve by using the computer software Mathematica, the values of constants obtained are as follows: Substituting Equation (14) in Equation (13), we get the solutions of Equation (10) as Case-II Substituting Equation (18) in Equation (13), we obtain the solutions of Equation (10) as Case-III Substituting Equation (22) in Equation (13), we get the solutions of Equation (10) as Case-IV Substituting Equation (26) in Equation (13), the solutions of Equation (10) are given as Substituting Equation (30) in Equation (13), we obtain the solutions of Equation (10) as

Case-VI
Substituting Equation (34) in Equation (13), the solutions of Equation (10) are given as

Results and discussion
Many researchers determined different types of solutions of KS equation by applying different techniques.
In this recent work, we have found new and more general exact travelling and solitary wave solutions, the important thing in this study is the trial solution of Equation (8)  In previous literature, many authors have been determined various types of solutions of KS equation such as elliptic, trigonometric, hyperbolic, rational functions including dark, bright, kink, anti-kink, periodic wave solutions with the help of modified truncated expansion method, dynamical system, bifurcation technique, backlund transformation, lie symmetry analysis, F-expansion method, improved subequation method, the ( G G )-expansion method [5][6][7][8][9][10][11][12][13][14][15][16]. But our obtained solutions have different structures in the shape of dark, bright, singular, combined, kink wave solitons, periodic solitary wave, traveling wave (see Figures 1-10).
In Figure 1   We can conclude that from the above comparison and detailed discussion, our obtained solutions are new and more generally which have not formulated before by other techniques. It is proved that our modified technique is fruitful, reliable, straight forward and effective to investigate other nonlinear evolution equations.

Conclusion
We successfully constructed some new exact travelling and solitary wave solutions of nonlinear KS equation by applying the modified mathematical technique. Our solutions are different and new from other researcher       found by using the different techniques before this work. Our new exact solutions obtained in the shape of dark solitons, bright solitons, travelling wave, solitary wave and periodic wave. These new solutions are more useful in the study of quantum plasma, optical fibres, dynamics of solitons, dynamics of fluid, problems of biomedical, mathematical physics, engineering and many other branches. The physical structure of new solutions shows the effectiveness and power of this technique. This research work completed by using the Mathematica. We can also apply this technique on other nonlinear evolution equations involves in optical fibre, Geo physics, mathematical physics, plasma physics, fluid dynamics, hydrodynamics, mechanics, mathematical biology, field of engineering and many other applied sciences.

Disclosure statement
No potential conflict of interest was reported by the authors.